P, Q, R are three (distinct) lattice points in the plane. Prove that if (PQ + QR)2 < 8 area PQR + 1, then P, Q, R are vertices of a square.
The integer coordinates imply that PQ2, QR2, and 2 area PQR are all integral. Hence (PQ + QR)2 ≤ 8 area PQR. Hence (PQ - QR)2 ≤ 8 area PQR - 4 PQ·QR. But 2 area PQR ≤ PQ·QR with equality if and only if PQR is a right-angle. Hence (PQ - QR)2 ≤ 0 with equality only if PQR is a right-angle. Hence PQ = QR and PQR is a right angle.
© John Scholes
12 Dec 1998